Finding the right hazard function for time‐to‐event modeling: A tutorial and Shiny application

Abstract

Parametric time‐to‐event analysis is an important pharmacometric method to predict the probability of an event up until a certain time as a function of covariates and/or drug exposure. Modeling is performed at the level of the hazard function describing the instantaneous rate of an event occurring at that timepoint. We give an overview of the parametric time‐to‐event analysis starting with graphical exploration by Kaplan–Meier plotting for the event data including censoring and nonparametric hazard estimators such as the kernel‐based visual hazard comparison for the underlying hazard. The most common hazard functions including the exponential, Gompertz, Weibull, log‐normal, log‐logistic, and circadian functions are described in detail. A Shiny application was developed to graphically guide the modeler which of the most common hazard functions presents a similar shape compared to the data in order to guide which hazard functions to test in the parametric time‐to‐event analysis. For the chosen hazard function(s), the Shiny application can additionally be used to explore corresponding parameter values to inform on suitable initial estimates for parametric modeling as well as on possible covariate or treatment relationships to certain parameters. Moreover, it can be used for the dissemination of results as well as communication, training, and workshops on time‐to‐event analysis. By guiding the modeler on which functions and what parameter values to test and compare as well as to assist in dissemination, the Shiny application developed here greatly supports the modeler in complicated parametric time‐to‐event modeling.

Seasonal influence on respiratory tract infection severity including COVID‐19 quantified through Markov Chain modeling

“…Nonlinear mixed effects modeling was performed using the Laplacian method in NONMEM 18 (version 7.5.0) assisted by PsN 19,20 (version 5.2.6). A reproducible workflow was developed for data handling and graphics throughout the trial, 21 which was performed in R 20 (version 4.0.4) through the RStudio 22 (version 1.4.1106) user interface.…”

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“…Parametric time-to-event (TTE) analysis was performed to analyze data. Exponential (Equation (1)), Weibull (Equation (2)), Gompterz (Equation (3)), and log-normal (Equation (4)) hazard distributions were evaluated as baseline hazard functions [ 30 ]. Initial estimates for the models were selected using the Shiny Application for parametric time-to-event analysis [ 30 ].…”

“…Exponential (Equation (1)), Weibull (Equation (2)), Gompterz (Equation (3)), and log-normal (Equation (4)) hazard distributions were evaluated as baseline hazard functions [ 30 ]. Initial estimates for the models were selected using the Shiny Application for parametric time-to-event analysis [ 30 ]. where t is time in days, λ is a scale parameter, α is a shape parameter, μ is the mean, σ is the standard deviation of a log-normal distribution, and Φ is the standard normal cumulative distribution function.…”

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“…12 As the authors did not report the number of patients in each gestational age group, the proportion of each group was extracted from the Kaplan-Meier analysis from the study and used to fit a time-to-event model. The time-to-event model consists of a survival function (Equation 1) and a Gompertz hazard function to describe the time of PDA closure in the absence of treatment with any drug (Equation 2) 13 :…”